Some analytic continuations of the Barnes zeta function in two and ...
Applied Mathematics and Computation 187 (2007) 141–152 www.elsevier.com/locate/amc
Some analytic continuations of the Barnes zeta function in two and higher dimensions E. Elizalde Consejo Superior de Investigaciones Cientı´ficas (ICE/CSIC), Institut d’Estudis Espacials de Catalunya (IEEC), Campus UAB, Facultat de Cie`ncies, Torre C5-Parell-2a Planta, E-08193 Bellaterra (Barcelona), Spain
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract Formulas for the analytic continuation of the Barnes zeta function, and some affine extensions thereof, in two and more dimensions, are constructed. The expressions are used to deal with determinants of multidimensional harmonic oscillators. An example is therewith obtained of the multiplicative anomaly (or defect), associated with the most common definition (due to Ray and Singer [D.B. Ray, Reidemeister torsion and the Laplacian on lens spaces, Adv. Math. 4 (1970) 109–126; D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145–201; D.B. Ray, I.M. Singer, Analytic torsion for complex manifolds, Ann. Math. 98 (1973) 154–177]) of determinant of a pseudodifferential operator admitting a zeta function [M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, in: Functional Analysis on the Eve of the 21st Century, vol. 1, 1995, pp. 173–197]. 2006 Elsevier Inc. All rights reserved. Keywords: Barnes zeta function; Hurwitz (or generalized) zeta; Determinant; Multiplicative anomaly (or defect); Quantum harmonic oscillators
1. Introduction A fundamental property shared by zeta functions is the existence of a reflection formula. For the Riemann zeta function it has the very simple form (for a useful and updated reference, see [26]): nð1 sÞ ¼ nðsÞ;
ð1:1Þ
nðsÞ : ps=2 Cðs=2ÞfðsÞ:
ð1:2Þ
being
E-mail address: [email protected] URL: http://www.ieec.fcr.es/english/recerca/ftc/eli/eli.htm 0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.110
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
For a generic zeta function, Z(s), we may write it as: Z(x s) = F(x, s)Z(s). It allows for its analytic continuation in a very easy way. This is, at least for the needs of ordinary cases, the main step of the zeta function regularization procedure in quantum physics. But the analytically continued expression thus obtained is most generally a series, of the same type as the initial one (that defines the zeta function in its domain of convergence on the complex plane, e.g., the half-plane to the left of the abscissa of convergence). Both series have a slow convergence behavior, of power type [1], which is extremely slow when approaching the abscissa of convergence. Some years ago, Chowla and Selberg (CS) found a formula, for the Epstein zeta function in the two-dimensional case [9], that yields exponentially quick convergence everywhere (not just in the reflected domain.) They were very proud of that formula. In Refs. [12–15], extensions to an arbitrary number of dimensions, both for the homogeneous (quadratic form) and non-homogeneous (quadratic plus affine form) cases were constructed. However, some of the new formulas (remarkably the ones corresponding to the zero-mass case, e.g., the original CS framework) were not explicit, since they involved solving a non-trivial recurrence. This was done in [16]. Aside from the quadratic case, which corresponds to the Epstein zeta function and its generalizations and which has a number of important applications in physics [3–5,11], the linear case is also most interesting (think just of a system of harmonic oscillators or a multidimensional oscillator, a very fundamental physical system in many applications). The most general linear zeta function studied to date is the Barnes’ one. Here again many explicit expressions for its analytic continuation are still missing, as well as for its derivative in the general case. We aim at filling a bit of this gap here. Some references by Srivastava and collaborators, related with this work are [6–8,27,29,30,35]. This paper is devoted to the case of the Barnes zeta function and extensions thereof (with a short incursion to the general affine case, a difficult one indeed). Some expressions are obtained for the derivatives of such linear zeta functions in quite general situations. Then, an application of these expressions, to deal with determinants [17] of multidimensional harmonic oscillators is considered in some detail. 2. The Barnes zeta function: dimension two case The Barnes zeta function, which corresponds to the linear case, turns out to be even more difficult to extend than the Epstein zeta function (quadratic case). This is easy to understand after some straightforward considerations. Back in the work of Riemann himself (and maybe even prior to this) it becomes clear that one of the main ingredients (if not the most essential) in his derivation of the properties of the zeta function is the use of the general Poisson summation formula, which states that, provided a given function, f, fulfills some quite general conditions, the sum of the values that f takes on an infinite, p-dimensional, integer lattice coincides with the sum of its Fourier transform, f~ , over the same lattice: 1 X
f ðn1 ; . . . ; np Þ ¼
n1 ;...;np ¼1
1 X
f~ ðn1 ; . . . ; np Þ:
ð2:1Þ
n1 ;...;np ¼1
Jacobi’s theta function identity for the h3 function may be considered a (one-dimensional) particular case of this formula, but in fact they are equivalent: anyone follows from the other. It becomes clear that both the conditions of good behavior at infinity demanded by the Poisson case, or the Gaussian functions appearing in Jacobi’s identity call for a quadratic dependence on the ni’s. The linear case lies outside the domain of applicability of these fundamental equations. As also does the case, very common in physical applications, of a truncated lattice (e.g., comprising the positive integers only). Put it in different terms, the expression of the Mellin inverse transform, ordinarily used for the analytic continuation of zeta functions, has a nice, Gaussian behavior for the quadratic case, over extense regions of the complex plane, but not so in the linear case. Anyway, linear and affine zeta functions turn out to be central in different physical applications [34,36]. Consider the Barnes zeta function in two dimensions [10] 1 X fB ðs; aj~ rÞ ¼ ða þ r1 n1 þ r2 n2 Þs ; Re s > 2; r1 ; r2 > 0; ð2:2Þ n1 ;n2 ¼0
and also the related zeta function
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
X
fðs; aj~ rÞ ¼
0
s
ða þ r1 n1 þ r2 n2 Þ ;
Re s > 2; r1 ; r2 > 0; r1 6¼ r2 ;
143
ð2:3Þ
n1 ;n2 2Z
where the prime means that the term with n1 = n2 = 0 is absent from the sum (actually, we could have defined the Barnes zeta function directly in this way, in order to allow for the particular case a = 0, but that would not have been the usual definition). Here we are implicitly assuming that the expressions in brackets never vanish, for any value of n1,n2. The two functions are related, provided we allow for the analytic continuation of the Barnes zeta function to negative values of the parameters r1,r2 (eventually, to complex values in general, since we are going to allow for the possibility r1, r2 2 C, in what follows). In fact: rÞ þ fB ðs; aj ~ rÞ þ fB ðs; ajðr1 ; r2 ÞÞ þ fB ðs; ajðr1 ; r2 ÞÞ fðs; aj~ rÞ ¼ fB ðs; aj~
2 X
s ½rs i fH ðs; a=r i Þ þ ðri Þ fH ðs; a=r i Þ;
ð2:4Þ
i¼1
being fH the Hurwitz zeta function. 2.1. An asymptotic expansion method We shall work, in a first approach, mainly at the level of asymptotic expansions. However, later we shall deal mostly with analytic, absolutely convergent series. Remember the asymptotic expansion for the Hurwitz zeta function 1 1 1s 1 s X Bk Cðs þ k 1Þ 1sk a þ a þ a fH ðs; aÞ ; a ! 1; ð2:5Þ s1 2 CðsÞ k! k¼2 where the Bk are Bernoulli numbers, and also the asymptotic expansion of its first derivative [18–20,25,21,22,2] 1 X a1s Bk Cðs þ k 1Þ 1sk a ½wðs þ k 1Þ wðsÞ f ðs; aÞ log a þ ; a ! 1: ð2:6Þ f0H ðs; aÞ H 2 CðsÞ k! ðs 1Þ k¼2 As starting point, we take the Barnes zeta function (2.2). It can be written as an infinite sum of Hurwitz zeta functions (absolutely convergent, for Re s big enough): 1 X a r1 rÞ ¼ rs f s; þ n fB ðs; aj~ H 2 r 2 r2 n¼0
1 r1s rs sB2 1s 1 r2 fH ðs 1; a=r1 Þ þ 1 fH ðs; a=r1 Þ þ r r2 fH ðs þ 1; a=r1 Þ þ OðsÞ: s1 2 2 1
ð2:7Þ
Performing at this instance the analytic continuation to s = 0, we obtain
r1 1 B2 r 2 r1 1 1 a r2 rÞ ¼ fH ð1; a=r1 Þ þ fH ð0; a=r1 Þ þ ¼ B2 ða=r1 Þ þ fB ð0; aj~ þ 2 2 2 r1 r2 2r1 2r2 12r1 a2 a 1 1 r1 r2 1 ¼ þ þ þ ; þ 2r1 r2 2 r1 r2 12r2 12r1 4
ð2:8Þ
which is the well known result for the Barnes zeta function at s = 0 in terms of generalized Bernoulli polynomials (see, e.g., Ref. [10]). A similar procedure can be used for obtaining the derivative of the Barnes zeta function, namely to take the derivative of the series in the first line of Eq. (2.7) (which is, again, absolutely convergent, for Re s big enough), to use then appropriate expansions for the derivatives of the Hurwitz zeta functions appearing there, and, finally, to analytically continue in s to the desired value, say s = 0. We now write down the successive expressions which follow from these steps, without further comment: 1 X a r1 0 0 s fB ðs; aj~ rÞ ¼ logðr2 ÞfB ðs; aj~ rÞ þ r2 fH s; þ n ; ð2:9Þ r2 r2 n¼0
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whereby 1s 1 X a r1 1 r1 a r1 a r1 f0H s; þ n f ðs 1;a=r Þ log þ n f s; þ n 1 H H r2 r2 r2 r2 r2 r2 ð1 sÞ2 r2 n¼0 n¼0 1sk 1 1 X Bk Cðs þ k 1Þ X a r1 þ ½wðs þ k 1Þ wðsÞ þ n CðsÞ r k! r2 2 k¼2 n¼0 1s 1þs 1 r1 sB2 r2 fH ðs 1;a=r1 Þ þ fH ðs þ 1;a=r1 Þ 2 r 2ð1 sÞ r1 2 ð1 sÞ 1s 1 r1 r2 log fH ðs 1;a=r1 Þ þ f0H ðs 1;a=r1 Þ ð1 sÞ r2 r1 s 1 r2 r2 þ log fH ðs; a=r1 Þ þ f0H ðs;a=r1 Þ 2 r1 r1 1þs sB2 r2 r2 0 þ log f ðs þ 1;a=r1 Þ þ fH ðs þ 1;a=r1 Þ 2 r1 r1 H 1sk 1 X Bk Cðs þ k 1Þ r1 þ ½wðs þ k 1Þ wðsÞ fH ðs þ k 1;a=r1 Þ þ OðsÞ: CðsÞ k! r2 k¼2
1 X
The analytic continuation to s = 0 yields the asymptotic series expansion (0 < r2 6 r1): 1 X a r1 r1 r1 r2 r1 1 a r2 0 fH 0; þ n B2 ða=r1 Þ 1 log log þ wða=r1 Þ þ log r 4 2r r 2r r 12r r r1 2 2 2 2 1 2 1 n¼0 1 a 1 r1 þ log C logð2pÞ f0H ð1; a=r1 Þ 2 r1 4 r2 1k 1 X Bk r1 þ fH ðk 1; a=r1 Þ: kðk 1Þ r2 k¼3
ð2:10Þ
ð2:11Þ
For the derivative at s = 0 of the Barnes zeta function in dimension two, we obtain the asymptotic series r1 1 1 r2 B2 ða=r1 Þ þ log Cða=r1 Þ logð2pÞ wða=r1 Þ 2 4 2r2 12r1 2kþ1 1 X r1 B2kþ2 r2 f0H ð1; a=r1 Þ þ fH ð2k þ 1; a=r1 Þ; r2 ð2k þ 1Þð2k þ 2Þ r1 k¼1
f0B ð0; aj~ rÞ fB ð0; aj~ rÞ log r1 þ
ð2:12Þ
in which the derivative of the Hurwitz zeta function at s = 1 can be written, in either of the two ways [21,22]: Z aða 1Þ a 1 0 logð2pÞ þ þ log CðaÞ fH ð1; aÞ ¼ 2 2 12 1 B2 ðaÞ 1 a 1 X B2kþ2 a2k log a : ð2:13Þ þ 2 2 4 8 k¼1 2kð2k þ 1Þð2k þ 2Þ This will require a specific determination of the logarithm (of the complex variables). Taking the second approach, we finally get 2 3a2 a a a r1 a r2 1 r2 f0B ð0; aj~ rÞ þ wða=r1 Þ log r1 log a þ 2r1 12r1 4 4r1 r2 2r2 2r1 r2 2r2 12r2 12r1 1 r 2k 1 a 1 r1 X B2kþ2 1 þ log C logð2pÞ þ 2 r1 4 r2 k¼1 2kð2k þ 1Þð2k þ 2Þ a 2kþ1 1 X B2kþ2 r2 þ fH ð2k þ 1; a=r1 Þ; r1 ; r2 < a: ð2:14Þ ð2k þ 1Þð2k þ 2Þ r1 k¼1
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
145
Observe also that the initial symmetry has been here explicitly broken (from the beginning) by the assumption jr2j 6 jr1 j, that has been crucial in obtaining the final formulas. This symmetry, once broken, seems impossible to recover at the end. 2.2. A rigorous procedure A different method starts from Hermite’s formula for the celebrated Hurwitz zeta function: Z 1 s=2 a1s as ða2 þ y 2 Þ sin½s arctanðy=aÞ þ þ2 : fH ðs; aÞ ¼ dy 2py e 1 s1 2 0
ð2:15Þ
This expression exhibits the singularity structure of fH(s, a) explicitly, the integral being an analytic function of s, "s 2 C (it is uniformly convergent for jsj 6 R, for any R > 0). Calling this integral I(s, a), it is immediate that I(s, a) ! 0, s ! 0, "a. For the derivative, we obtain, 1s Z 1 s=2 a1s a as ða2 þ y 2 Þ logða2 þ y 2 Þ sin½s arctanðy=aÞ 0 þ fH ðs; aÞ ¼ dy log a e2py 1 s1 2 ðs 1Þ2 0 Z 1 s=2 ða2 þ y 2 Þ arctanðy=aÞ cos½s arctanðy=aÞ þ2 : ð2:16Þ dy e2py 1 0 Calling the new integrals I1(s, a) and I2(s, a), respectively, we get I 1 ðs; aÞ ! 0; I 2 ðs; aÞ ! 2a
s ! 0; Z
1
dx 0
arctan x 1 1 þ Oða2 Þ ; ¼ e2pax 1 12a
s ! 0;
ð2:17Þ
and it follows that k 1 1 1X ð1Þ ð2kÞ!fð2k þ 2Þ : f0H ð0; aÞ a þ a log a þ 2kþ1 2 p k¼0 ð2paÞ
ð2:18Þ
Recall that the Barnes zeta function is written in terms of the Hurwitz zeta function, as given by the first of Eqs. (2.7) above. To obtain the derivative of the Barnes zeta function, however, we will proceed as before: we first work in terms of a general s in order to be able to analytically continue the expression to the desired Umgebung of s = 0, expand then in Taylor series around s = 0, and finally take the derivative at the point s = 0. Being more precise, we first expand fH(s, a) around s = 0, what yields fB ðs; aj~ rÞ ¼
rs r1 rs rs r2 1 fH ðs 1; a=r1 Þ þ 1 fH ðs; a=r1 Þ þ 1 sf ð1 þ s; a=r1 Þ þ Rðs; aj~ rÞ; s 1 r2 2 12 r1 H
ð2:19Þ
where the remainder term can be expressed in the following alternative ways (other possibilities may exist, too) "Z 8 2 # 1 1 > P arctan x dx 1 a r1 r1s > 1s 1 > ðn þ a=r1 Þ þ n þ Oðs2 Þ; > < ¼ 2s r2 exp½2pðr1 =r2 Þxðn þ a=r1 Þ 1 24 r2 r2 n¼0 0 Rðs; aj~ rÞ > 2kþ1 1 > srs P > r2 > ð1Þk ð2kÞ!fð2k þ 2Þ 2pr fH ðs þ 2k þ 1; a=r1 Þ þ Oðs2 Þ: : p1 1 k¼1
ð2:20Þ From here it is immediate to obtain, at s = 0, the formula (cf. Eq. (2.8)): r1 1 1 a r2 fB ð0; aj~ rÞ ¼ B2 ða=r1 Þ þ : þ 2 2 r1 2r2 12r1
ð2:21Þ
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
And from Eqs. (2.19), (2.20), taking the derivative with respect to s, at s = 0, we finally arrive to the very simple expression rÞ ¼ fB ð0; aj~ rÞ log r1 þ f0B ð0; aj~ where
Rðaj~ rÞ
r1 0 f ð1; a=r1 Þ þ Rðaj~ rÞ; r2 H
"Z 8 1 > P > 2r1 > ðn þ a=r1 Þ > < ¼ r2 > > > > : p1
r1 1 1 r2 B2 ða=r1 Þ þ log Cða=r1 Þ logð2pÞ wða=r1 Þ 2 4 2r2 12r1 ð2:22Þ
2 # arctan x dx 1 a r1 þ n ; exp ½2pðr1 =r2 Þxðn þ a=r1 Þ 1 24 r2 r2 n¼0 0 2kþ1 1 P k r2 ð1Þ ð2kÞ!fð2k þ 2Þ 2pr fH ð2k þ 1; a=r1 Þ: 1 1
ð2:23Þ
k¼1
Observe that, here, analyticity has been rigorously preserved at every step of the calculation. Those in Eq. (2.23) are two possible expressions for the remainder term Rða j ~ rÞ. The first of them is valid for 0 < r2 6 r1. It is very quickly convergent in this region and, therefore, very appropriate for numerical computation. Concerning this extreme, we note that, calling Z 1 1 X arctan x dx 1 ða þ bnÞ2 ; f ða; bÞ 2 ða þ bnÞ ð2:24Þ exp½2pða þ bnÞx 1 24 0 n¼0 one gets immediately (even with a pocket calculator) f ð1; 1Þ ¼ 0:0028;
f ð1; 10Þ ¼ 0:0023;
f ði; 1Þ ¼ 3:6 104 þ i0:084;
f ð10; 1Þ ¼ 1:5 105
f ð10i; 1Þ ¼ 2 106 þ i0:008;
f ð10; iÞ ¼ 3:3 105 þ i1:4 105 :
ð2:25Þ
Observe the very quick convergence of the integral expression, even for imaginary values of the argument. The second form of the remainder, Eq. (2.23), is an explicit asymptotic series for r2 2pr1. It coincides term by term with the asymptotic expansion obtained before. In fact, this second representation is exactly the same of the preceding subsection (cf. Eq. (2.12)). Notice once more that the initial symmetry under interchange of these two parameters has been explicitly broken by this assumption that permits the whole calculation to be carried out. Moreover, by treating apart (if necessary) a finite number of first terms in the original expression, it is easy to artificially increase the value of a and, with this, to considerably augment the speed and accuracy of the numerical results, in the first of expressions (2.23). 2.3. Analytic continuation to arbitrary values of r1, r2 Taking as starting point for the analytic continuation the formula: Z 1 1 1 X 1 X s ða þ r1 n1 þ r2 n2 Þ ¼ dt ts1 eðaþr1 n1 þr2 n2 Þt CðsÞ 0 n1 ;n2 ¼0 n1 ;n2 ¼0 Z 1 1 eat ; a; r1 ; r2 > 0; Re s > 2; ¼ dt ts1 CðsÞ 0 ð1 er1 t Þð1 er2 t Þ
ð2:26Þ
it is plain that it remains valid without change as long as the real parts of a, r1, r2 are all positive. It is also clear that in the complex planes corresponding to the parameters r1, r2, as soon as any of them gets purely imaginary, say r1 ¼ jr1 jeip=2 ¼ ijr1 j;
ð2:27Þ
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
than an infinite number of poles are created in the integrand, with X s1 1 X
1 eat 2pk s1 Re s t e2pka=jr1 j 1 e2pkr2 =jr1 j D1 ; ¼ r1 t Þð1 er2 t Þ jr ð1 e j 1 k¼1 r -poles
147
ð2:28Þ
1
which is an absolutely convergent series that will provide an additional contribution, 2pD1, in the analytic continuation process when, equivalently, the integration path for the integrand variable t (originally over the positive real axis) is deformed beyond the positive imaginary axis, in order to reach the possibility of setting Re r1 < 0 in the integral formula for the new integration path. Complementing this procedure—namely, just for one of the two parameters ri—with a formula for the analytic continuation of the Hurwitz zeta function fH(s, a) in the parameter a, we are done. This last can be read off from the usual tables, with rigorous prescriptions and domains of validity. Thus, the final formula for the analytic continuation in the parameter r1 has the form: Z 1 X 1 eas s þ 2pD1 ; Re a; Re r2 > 0; Re r1 < 0; ða þ r1 n1 þ r2 n2 Þ ¼ dsss1 r CðsÞ R ð1 e 1 s Þð1 er2 s Þ n ;n ¼0 1
2
ð2:29Þ being R a straight-line in the second quadrant of the r1-complex plane, starting at the origin. 2.4. Barnes’ related zeta function in two dimensions For the other, Barnes’ related, two-dimensional zeta function considered at the beginning, Eq. (2.3), we get 4 2 X X 1 a 1 a a a þ f0 ð0; aj~ rÞ ¼ f0B ð0;aj~ ra Þ þ log ri þ logðri Þ log C log C þ logð2pÞ ; 2 ri 2 ri ri ri a¼1 i¼1 ð2:30Þ where ~ r1 ¼ ðr1 ; r2 Þ;
~ r2 ¼ ðr1 ; r2 Þ;
~ r3 ¼ ðr1 ; r2 Þ;
~ r4 ¼ ðr1 ; r2 Þ:
ð2:31Þ
Note that this evaluation involves an analytic continuation on the values of r1 and r2—which are strictly positive in the case of the Barnes zeta function—to negative and, in general, complex values r1, r2 2 C. In this way we try to elude the unsurmountable problems that a direct interpretation of the sum in Eq. (2.3) poses. In particular, when r1 = r2 it develops an infinite number of zero and constant modes for an infinite number of possible values of the constants, what renders a direct interpretation of the series extremely problematic. After the analytic continuation performed here from the Barnes zeta function—together with a natural transportation of the negative signs of the indices ni to the parameters ri—the above limit is obtained in the final expressions, although with some severe restrictions (see below). Substituting the expressions (2.22) and (2.23) above into Eq. (2.30), after some long but straightforward computation, we obtain a a api a p 0 ð2:32Þ f ð0; aj~ rÞ ¼ log C log i þ log 2p: C þ r2 r2 r2 r2 2 Important considerations are here in order. (i) Observe, to begin with, that the last one is a finite expression and not an asymptotic series expansion as for the case of the Barnes zeta function. In fact, it has happened that the terms of the asymptotic expansions canceled one by one when performing the sum over all the ~ r 0 s. (ii) Second, note that the breaking of the initial symmetry under interchange of r1 and r2 in dealing with the analytic continuation (valid for r2 6 r1), has finally resulted in the disappearance of one of the two parameters from the end result. Remarkable is the fact, however, that the final formula actually preserves the modular invariance of the initial one, under the parameter change on the other variable, here r2 a ! a þ kr2 ;
k 2 Z;
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
since it transforms as f0 ð0; aj~ rÞ ! f0 ð0; aj~ rÞ þ 2pk; so that the modular symmetry of a with respect to r2 is preserved. What has happened to the r1 dependence? It has disappeared completely, the reason for this being the following. The formula for the Barnes zeta function for positive r1 and r2 obtained in the preceding section cannot be analytically continued to both r1 and r2 negative, because, for any value of s (and a, r1, r2 fixed), once we break the symmetry and do the analytic continuation say in r2, then the analytic continuation in r1 is restricted to the only possibility for n1 of being zero. In fact, in other words, when we try to pull out of the r1 real axis, for any value of n1 5 0 we encounter a pole of the initial zeta function, for Im r1 arbitrarily small—as these poles form a dense set in the complex r1plane—taking n2 big enough. Now, it is immediate to see that one can interpret the final formula (2.32) as the true analytic continuation for the restriction n1 = 0, namely: 1 X
f0 ð0; ajð0; r2 ÞÞ ¼ f0 ð0; ajð0; r2 ÞÞ ¼ ð2:32Þ:
ð2:33Þ
1
Let us quote also the following result, that we have obtained in two different ways: by analytically continuing in the factor 1 of n1 to 1 or, alternatively, on a and i to a and i, simultaneously. We get 1 1 X X
ða þ n1 þ in2 Þ
s
aið1 log aÞ
n1 ¼1 n2 ¼0
ipa p pa i ½wðaÞ wðaÞ 2 12 2 12
1 log½CðaÞCðaÞ log CðaiÞ 2 k 1 X ð1Þ B2kþ2 ½fH ð2k þ 1; aÞ fH ð2k þ 1; aÞ: þi ð2k þ 1Þð2k þ 2Þ k¼1
þ
ð2:34Þ
Finally, these formulas give us immediately the values of the determinants of the corresponding operators (say A) in each case [31–33]: rÞ ¼ exp f0A ð0; aj~ rÞ : detf Aðaj~ This issue will be further analyzed in the next section. 3. Barnes zeta functions in arbitrary dimensions There is no obstruction, in principle, to using the same procedure with general Barnes zeta functions in any dimension, d, and getting in this way corresponding formulas of the same type (involving multi-sums). The method will be described in this section. Using once more Hermite’s formula, we readily obtain the following recurrence for the Barnes zeta function in d dimensions ðdÞ
fB ðs; aj~ rÞ ¼
1 X
s
ða þ~ rd ~ nd Þ ;
Re s > d;
ð3:1Þ
0 ~ nd ¼~
in terms of corresponding ones in d 1 dimensions ðdÞ
fB ðs; aj~ rÞ ¼
r1 1 ðd1Þ ðd1Þ d fB ðs 1; aj~ rd1 Þ þ fB ðs; aj~ rd1 Þ 2 s1 Z 1 s=2 1 X ð1 þ x2 Þ sinðs arctan xÞ 1s 1 ða þ~ rd1 ~ nd1 Þ dx þ rd 1 2prd ðaþ~ rd1 ~ nd1 Þx e 1 0 ~ 0 n ¼~ d1
ð3:2Þ
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149
Proceeding as in the section before, after some work we obtain the following analytic expansion of ðdÞ fB ðs; aj~ rÞ around s = 0: ðdÞ
fB ðs; aj~ rÞ ¼
½d=21 2kþ1 r1 1 ðd1Þ s X ðd1Þ k rd d fB ðs 1; aj~ rd1 Þ þ fB ðs; aj~ rd1 Þ þ ð1Þ 2 2p k¼0 s1 2p 1 X ðd1Þ 1s ð2kÞ!fð2k þ 2ÞfB ð2k þ s þ 1; aj~ rd1 Þ þ sr1 ða þ~ rd1 ~ nd1 Þ d ~ 0 nd1 ¼~
Z
1
dx 0
arctan x 1 ðaþ~ rd1 ~ nd1 Þx
e2prd
1
½d=21 X
ð1Þ
k¼0
k
r 2kþ2 d
2p
ð2kÞ!ða þ~ rd1 ~ nd1 Þ
# 2ðkþ1Þ
þ Oðs2 Þ: ð3:3Þ
Here [d/2 1] means integer part of d/2 1. The added and subtracted terms correspond to the poles of the d-dimensional Barnes zeta function. The convergence properties of the last term, i.e., the integral minus the subtracted terms, are as good for general d as for the case d = 2 (see the preceding section). 3.1. Derivatives of higher dimensional Barnes zeta functions We can now take the derivative at s = 0, with the following result: ðdÞ 0
1 ðd1Þ 0 ð1; aj~ rd1 Þ þ fB ð0; aj~ rd1 Þ 2 ½d=21 h i r 2kþ1 1 X d ðd1Þ þ ð1Þk ð2kÞ!fð2k þ 2ÞFP fB ð2k þ 1; aj~ rd1 Þ þ Rd ð0; aj~ rÞ; 2p k¼0 2p ðd1Þ
fB ð0; aj~ rÞ ¼ r1 d fB
ðd1Þ 0
ð1; aj~ rd1 Þ r1 d fB
ð3:4Þ
where FP[ ] means ‘finite part of’, and for the remainder term we have, as before, the two different expressions: "Z 8 1 1 P arctan x > > > ða þ~ rd1 ~ nd1 Þ dx 2pr1 ðaþ~r ~n Þx ¼ r1 > d > d1 d1 d > 1 e 0 ~ 0 nd1 ¼~ > > # > < ½d=21 2kþ2 P k rd 2ðkþ1Þ ð3:5Þ Rd ð0; aj~ rÞ ð1Þ ð2kÞ!ða þ~ rd1 ~ nd1 Þ ; > > 2p k¼0 > > > 2kþ1 > 1 > P > ðd1Þ k rd 1 > ð1Þ ð2kÞ!fð2k þ 2ÞfB ð2k þ 1; aj~ rd1 Þ: : 2p 2p k¼½d=2 ðdÞ 0
Observe that in order 0to obtain fB ð0; aj~ rÞ we need two derivatives of the d 1 zeta function, namely ðd1Þ 0 ðd1Þ fB ð0; aj~ rd1 Þ and fB ð1; aj~ rd1 Þ. The derivative at s = 1 can be calculated exactly in the same way as the one at s = 0, just by performing first the expansion of the Barnes zeta function around s = 1. For the case d = 2 (that is, for the first derivative needed in order to initiate the recurrence), the result is the following: r1 r1 0 ð2Þ0 ð2Þ fB ð1; aj~ rÞ ¼ r1 log r1 fB ð1; aj~ rÞ þ r1 f ð2; a=r1 Þ f ð2; a=r1 Þ 4r2 H 2r2 H 1 r2 r2 0 þ f0H ð1; a=r1 Þ f ð0; a=r1 Þ f ð0; a=r1 Þ þ R1 ð1; aj~ rÞ; ð3:6Þ 2 12r1 H 12r1 H the remainder term R1 ð1; aj~ rÞ being finite and given by the asymptotic expansion: 2kþ1 1 r1 X r2 k R1 ð1; aj~ rÞ ¼ ð1Þ ð2k 1Þ!fð2k þ 2Þ fH ð2k; a=r1 Þ; p k¼1 2pr1
ð3:7Þ
or by the corresponding expression as the difference of an integral and the term k = 0 of the series (same situation as above).
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With this we close the circle and, we observe that this procedure allows for the calculation of any derivative of the Barnes zeta function in d dimensions in a recursive way and with a highly accurate numerical determination of the remainder terms. 3.2. Application to the multidimensional harmonic oscillator: determinants and the multiplicative anomaly (or defect) Let us consider the quantum harmonic oscillator in d dimensions, with angular frequencies (x1, . . . , xd). The eigenvalues read þ b; kn ¼ n x
n ðn1 ; . . . ; nd Þ;
ðx1 ; . . . ; xd Þ; x
b¼
d 1X xk 2 k¼1
ð3:8Þ
whose poles are to be found at the points and the related zeta function is the Barnes one, fd ðs; b j xÞ, s = k(k = d, d 1, . . . , 1). Their corresponding residua can be expressed in terms of generalized Bernoulli polyðdÞ defined by nomials Bdk ðbjxÞ, Qd
td eat
i¼1 ð1
ebi t Þ
1 ¼ Qd
1 X
i¼1 bi n¼0
n
BðdÞ n ðajbi Þ
ðtÞ : n!
ð3:9Þ
The residua of the Barnes zeta function are ¼ Re sfd ðk; bjxÞ
ð1Þ
dþk
ðk 1Þ!ðd kÞ!
Qd
j¼1 xj
ðdÞ
Bdk ðbjxÞ;
k ¼ d; d 1; . . . :
ð3:10Þ
At issue will be here the calculation, through the corresponding zeta function, of the determinant of the harmonic oscillator Hamiltonian in any number, d, of dimensions. As discussed before, in Quantum Field Theory the zeta function method is unchallenged at the one-loop level, where it is rigorously defined and where many calculations reduce basically to the computation of determinants. It is interesting to note, however, that for the determinants defined by means of the zeta function [31–33], detf A :¼ exp½f0A ð0Þ;
ð3:11Þ
the question of the non-commutative or multiplicative anomaly [28] appears (see, for instance [23,24]): the determinant of the product of two pseudodifferential operators (admitting both and their product a zeta function and corresponding zeta-determinants) is generically not equal to the product of the determinants. The logarithm of the quotient of the two results is termed the non-commutative (albeit it is actually non-zero in many cases where the two operators do commute, as we are going to see right now), or multiplicative anomaly.1 Using the formulas of the preceding sections, we are able to obtain the determinant of the harmonic oscillator Hamiltonian in d dimensions. We can then compare the result with the one obtained by making use of the celebrated formula of Wodzicki for the multiplicative anomaly: h i res ln Ab Ba 2 detf ðABÞ ; ð3:12Þ aðA; BÞ :¼ log ¼ detf ðAÞdetf ðBÞ 2abða þ bÞ where a > 0 and b > 0 are the orders of A and B respectively, while res is the Wodzicki or non-commutative residue (that is, the unique trace, up to multiplication by a constant, in the class of pseudodifferential operators) [37]. This formula is, in principle, not guaranteed to provide the right answer for the present situation, which involves non-compact operators. 1
Isadore Singer calls it a determinant defect. I thank him for illuminating discussions at his office at MIT, and for the very good tea.
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
151
Let V a constant potential, H the Hamiltonian operator corresponding to the free quantum harmonic oscillator in d dimensions, and as second operator let us take HV :¼ H + V (which actually commutes with H). For the multiplicative anomaly, we obtain ½d=2 ðdÞ 2k ð1Þd X ½c þ wðd 2kÞB2k ðbjxÞ aðH ; H V Þ ¼ Qd V : ð2kÞ!ðd 2kÞ! 2 j¼1 xj k¼1
ð3:13Þ
Here the generalized Bernoulli polynomials of odd order vanish and thus they can be dismissed from the sum. In spite of the manifold being non-compact, we are able to confirm the validity of the Wodzicki formula in the case under study, since exactly the same result, Eq. (3.13), is obtained by making use of Eq. (3.12). On the other hand, the remaining generalized Bernoulli polynomials are never zero, in fact d 1 X ðdÞ ¼ 1; BðdÞ B0 ðbjxÞ ðbj xÞ ¼ x2 ; 2 12 i¼1 i " # d X 1 7 X ðdÞ 4 2 2 ¼ x þ xi xj ; B4 ðbjxÞ ð3:14Þ 24 10 i¼1 i i<j " # d 5 31 X 7 X 4 2 X 2 2 2 ðdÞ 6 ¼ x þ xx þ x x x ; ... B6 ðbjxÞ 96 70 i¼1 i 10 i6¼j i j i<j
Some analytic continuations of the Barnes zeta function in two and higher dimensions E. Elizalde Consejo Superior de Investigaciones Cientı´ficas (ICE/CSIC), Institut d’Estudis Espacials de Catalunya (IEEC), Campus UAB, Facultat de Cie`ncies, Torre C5-Parell-2a Planta, E-08193 Bellaterra (Barcelona), Spain
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract Formulas for the analytic continuation of the Barnes zeta function, and some affine extensions thereof, in two and more dimensions, are constructed. The expressions are used to deal with determinants of multidimensional harmonic oscillators. An example is therewith obtained of the multiplicative anomaly (or defect), associated with the most common definition (due to Ray and Singer [D.B. Ray, Reidemeister torsion and the Laplacian on lens spaces, Adv. Math. 4 (1970) 109–126; D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145–201; D.B. Ray, I.M. Singer, Analytic torsion for complex manifolds, Ann. Math. 98 (1973) 154–177]) of determinant of a pseudodifferential operator admitting a zeta function [M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, in: Functional Analysis on the Eve of the 21st Century, vol. 1, 1995, pp. 173–197]. 2006 Elsevier Inc. All rights reserved. Keywords: Barnes zeta function; Hurwitz (or generalized) zeta; Determinant; Multiplicative anomaly (or defect); Quantum harmonic oscillators
1. Introduction A fundamental property shared by zeta functions is the existence of a reflection formula. For the Riemann zeta function it has the very simple form (for a useful and updated reference, see [26]): nð1 sÞ ¼ nðsÞ;
ð1:1Þ
nðsÞ : ps=2 Cðs=2ÞfðsÞ:
ð1:2Þ
being
E-mail address: [email protected] URL: http://www.ieec.fcr.es/english/recerca/ftc/eli/eli.htm 0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.110
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For a generic zeta function, Z(s), we may write it as: Z(x s) = F(x, s)Z(s). It allows for its analytic continuation in a very easy way. This is, at least for the needs of ordinary cases, the main step of the zeta function regularization procedure in quantum physics. But the analytically continued expression thus obtained is most generally a series, of the same type as the initial one (that defines the zeta function in its domain of convergence on the complex plane, e.g., the half-plane to the left of the abscissa of convergence). Both series have a slow convergence behavior, of power type [1], which is extremely slow when approaching the abscissa of convergence. Some years ago, Chowla and Selberg (CS) found a formula, for the Epstein zeta function in the two-dimensional case [9], that yields exponentially quick convergence everywhere (not just in the reflected domain.) They were very proud of that formula. In Refs. [12–15], extensions to an arbitrary number of dimensions, both for the homogeneous (quadratic form) and non-homogeneous (quadratic plus affine form) cases were constructed. However, some of the new formulas (remarkably the ones corresponding to the zero-mass case, e.g., the original CS framework) were not explicit, since they involved solving a non-trivial recurrence. This was done in [16]. Aside from the quadratic case, which corresponds to the Epstein zeta function and its generalizations and which has a number of important applications in physics [3–5,11], the linear case is also most interesting (think just of a system of harmonic oscillators or a multidimensional oscillator, a very fundamental physical system in many applications). The most general linear zeta function studied to date is the Barnes’ one. Here again many explicit expressions for its analytic continuation are still missing, as well as for its derivative in the general case. We aim at filling a bit of this gap here. Some references by Srivastava and collaborators, related with this work are [6–8,27,29,30,35]. This paper is devoted to the case of the Barnes zeta function and extensions thereof (with a short incursion to the general affine case, a difficult one indeed). Some expressions are obtained for the derivatives of such linear zeta functions in quite general situations. Then, an application of these expressions, to deal with determinants [17] of multidimensional harmonic oscillators is considered in some detail. 2. The Barnes zeta function: dimension two case The Barnes zeta function, which corresponds to the linear case, turns out to be even more difficult to extend than the Epstein zeta function (quadratic case). This is easy to understand after some straightforward considerations. Back in the work of Riemann himself (and maybe even prior to this) it becomes clear that one of the main ingredients (if not the most essential) in his derivation of the properties of the zeta function is the use of the general Poisson summation formula, which states that, provided a given function, f, fulfills some quite general conditions, the sum of the values that f takes on an infinite, p-dimensional, integer lattice coincides with the sum of its Fourier transform, f~ , over the same lattice: 1 X
f ðn1 ; . . . ; np Þ ¼
n1 ;...;np ¼1
1 X
f~ ðn1 ; . . . ; np Þ:
ð2:1Þ
n1 ;...;np ¼1
Jacobi’s theta function identity for the h3 function may be considered a (one-dimensional) particular case of this formula, but in fact they are equivalent: anyone follows from the other. It becomes clear that both the conditions of good behavior at infinity demanded by the Poisson case, or the Gaussian functions appearing in Jacobi’s identity call for a quadratic dependence on the ni’s. The linear case lies outside the domain of applicability of these fundamental equations. As also does the case, very common in physical applications, of a truncated lattice (e.g., comprising the positive integers only). Put it in different terms, the expression of the Mellin inverse transform, ordinarily used for the analytic continuation of zeta functions, has a nice, Gaussian behavior for the quadratic case, over extense regions of the complex plane, but not so in the linear case. Anyway, linear and affine zeta functions turn out to be central in different physical applications [34,36]. Consider the Barnes zeta function in two dimensions [10] 1 X fB ðs; aj~ rÞ ¼ ða þ r1 n1 þ r2 n2 Þs ; Re s > 2; r1 ; r2 > 0; ð2:2Þ n1 ;n2 ¼0
and also the related zeta function
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
X
fðs; aj~ rÞ ¼
0
s
ða þ r1 n1 þ r2 n2 Þ ;
Re s > 2; r1 ; r2 > 0; r1 6¼ r2 ;
143
ð2:3Þ
n1 ;n2 2Z
where the prime means that the term with n1 = n2 = 0 is absent from the sum (actually, we could have defined the Barnes zeta function directly in this way, in order to allow for the particular case a = 0, but that would not have been the usual definition). Here we are implicitly assuming that the expressions in brackets never vanish, for any value of n1,n2. The two functions are related, provided we allow for the analytic continuation of the Barnes zeta function to negative values of the parameters r1,r2 (eventually, to complex values in general, since we are going to allow for the possibility r1, r2 2 C, in what follows). In fact: rÞ þ fB ðs; aj ~ rÞ þ fB ðs; ajðr1 ; r2 ÞÞ þ fB ðs; ajðr1 ; r2 ÞÞ fðs; aj~ rÞ ¼ fB ðs; aj~
2 X
s ½rs i fH ðs; a=r i Þ þ ðri Þ fH ðs; a=r i Þ;
ð2:4Þ
i¼1
being fH the Hurwitz zeta function. 2.1. An asymptotic expansion method We shall work, in a first approach, mainly at the level of asymptotic expansions. However, later we shall deal mostly with analytic, absolutely convergent series. Remember the asymptotic expansion for the Hurwitz zeta function 1 1 1s 1 s X Bk Cðs þ k 1Þ 1sk a þ a þ a fH ðs; aÞ ; a ! 1; ð2:5Þ s1 2 CðsÞ k! k¼2 where the Bk are Bernoulli numbers, and also the asymptotic expansion of its first derivative [18–20,25,21,22,2] 1 X a1s Bk Cðs þ k 1Þ 1sk a ½wðs þ k 1Þ wðsÞ f ðs; aÞ log a þ ; a ! 1: ð2:6Þ f0H ðs; aÞ H 2 CðsÞ k! ðs 1Þ k¼2 As starting point, we take the Barnes zeta function (2.2). It can be written as an infinite sum of Hurwitz zeta functions (absolutely convergent, for Re s big enough): 1 X a r1 rÞ ¼ rs f s; þ n fB ðs; aj~ H 2 r 2 r2 n¼0
1 r1s rs sB2 1s 1 r2 fH ðs 1; a=r1 Þ þ 1 fH ðs; a=r1 Þ þ r r2 fH ðs þ 1; a=r1 Þ þ OðsÞ: s1 2 2 1
ð2:7Þ
Performing at this instance the analytic continuation to s = 0, we obtain
r1 1 B2 r 2 r1 1 1 a r2 rÞ ¼ fH ð1; a=r1 Þ þ fH ð0; a=r1 Þ þ ¼ B2 ða=r1 Þ þ fB ð0; aj~ þ 2 2 2 r1 r2 2r1 2r2 12r1 a2 a 1 1 r1 r2 1 ¼ þ þ þ ; þ 2r1 r2 2 r1 r2 12r2 12r1 4
ð2:8Þ
which is the well known result for the Barnes zeta function at s = 0 in terms of generalized Bernoulli polynomials (see, e.g., Ref. [10]). A similar procedure can be used for obtaining the derivative of the Barnes zeta function, namely to take the derivative of the series in the first line of Eq. (2.7) (which is, again, absolutely convergent, for Re s big enough), to use then appropriate expansions for the derivatives of the Hurwitz zeta functions appearing there, and, finally, to analytically continue in s to the desired value, say s = 0. We now write down the successive expressions which follow from these steps, without further comment: 1 X a r1 0 0 s fB ðs; aj~ rÞ ¼ logðr2 ÞfB ðs; aj~ rÞ þ r2 fH s; þ n ; ð2:9Þ r2 r2 n¼0
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
whereby 1s 1 X a r1 1 r1 a r1 a r1 f0H s; þ n f ðs 1;a=r Þ log þ n f s; þ n 1 H H r2 r2 r2 r2 r2 r2 ð1 sÞ2 r2 n¼0 n¼0 1sk 1 1 X Bk Cðs þ k 1Þ X a r1 þ ½wðs þ k 1Þ wðsÞ þ n CðsÞ r k! r2 2 k¼2 n¼0 1s 1þs 1 r1 sB2 r2 fH ðs 1;a=r1 Þ þ fH ðs þ 1;a=r1 Þ 2 r 2ð1 sÞ r1 2 ð1 sÞ 1s 1 r1 r2 log fH ðs 1;a=r1 Þ þ f0H ðs 1;a=r1 Þ ð1 sÞ r2 r1 s 1 r2 r2 þ log fH ðs; a=r1 Þ þ f0H ðs;a=r1 Þ 2 r1 r1 1þs sB2 r2 r2 0 þ log f ðs þ 1;a=r1 Þ þ fH ðs þ 1;a=r1 Þ 2 r1 r1 H 1sk 1 X Bk Cðs þ k 1Þ r1 þ ½wðs þ k 1Þ wðsÞ fH ðs þ k 1;a=r1 Þ þ OðsÞ: CðsÞ k! r2 k¼2
1 X
The analytic continuation to s = 0 yields the asymptotic series expansion (0 < r2 6 r1): 1 X a r1 r1 r1 r2 r1 1 a r2 0 fH 0; þ n B2 ða=r1 Þ 1 log log þ wða=r1 Þ þ log r 4 2r r 2r r 12r r r1 2 2 2 2 1 2 1 n¼0 1 a 1 r1 þ log C logð2pÞ f0H ð1; a=r1 Þ 2 r1 4 r2 1k 1 X Bk r1 þ fH ðk 1; a=r1 Þ: kðk 1Þ r2 k¼3
ð2:10Þ
ð2:11Þ
For the derivative at s = 0 of the Barnes zeta function in dimension two, we obtain the asymptotic series r1 1 1 r2 B2 ða=r1 Þ þ log Cða=r1 Þ logð2pÞ wða=r1 Þ 2 4 2r2 12r1 2kþ1 1 X r1 B2kþ2 r2 f0H ð1; a=r1 Þ þ fH ð2k þ 1; a=r1 Þ; r2 ð2k þ 1Þð2k þ 2Þ r1 k¼1
f0B ð0; aj~ rÞ fB ð0; aj~ rÞ log r1 þ
ð2:12Þ
in which the derivative of the Hurwitz zeta function at s = 1 can be written, in either of the two ways [21,22]: Z aða 1Þ a 1 0 logð2pÞ þ þ log CðaÞ fH ð1; aÞ ¼ 2 2 12 1 B2 ðaÞ 1 a 1 X B2kþ2 a2k log a : ð2:13Þ þ 2 2 4 8 k¼1 2kð2k þ 1Þð2k þ 2Þ This will require a specific determination of the logarithm (of the complex variables). Taking the second approach, we finally get 2 3a2 a a a r1 a r2 1 r2 f0B ð0; aj~ rÞ þ wða=r1 Þ log r1 log a þ 2r1 12r1 4 4r1 r2 2r2 2r1 r2 2r2 12r2 12r1 1 r 2k 1 a 1 r1 X B2kþ2 1 þ log C logð2pÞ þ 2 r1 4 r2 k¼1 2kð2k þ 1Þð2k þ 2Þ a 2kþ1 1 X B2kþ2 r2 þ fH ð2k þ 1; a=r1 Þ; r1 ; r2 < a: ð2:14Þ ð2k þ 1Þð2k þ 2Þ r1 k¼1
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
145
Observe also that the initial symmetry has been here explicitly broken (from the beginning) by the assumption jr2j 6 jr1 j, that has been crucial in obtaining the final formulas. This symmetry, once broken, seems impossible to recover at the end. 2.2. A rigorous procedure A different method starts from Hermite’s formula for the celebrated Hurwitz zeta function: Z 1 s=2 a1s as ða2 þ y 2 Þ sin½s arctanðy=aÞ þ þ2 : fH ðs; aÞ ¼ dy 2py e 1 s1 2 0
ð2:15Þ
This expression exhibits the singularity structure of fH(s, a) explicitly, the integral being an analytic function of s, "s 2 C (it is uniformly convergent for jsj 6 R, for any R > 0). Calling this integral I(s, a), it is immediate that I(s, a) ! 0, s ! 0, "a. For the derivative, we obtain, 1s Z 1 s=2 a1s a as ða2 þ y 2 Þ logða2 þ y 2 Þ sin½s arctanðy=aÞ 0 þ fH ðs; aÞ ¼ dy log a e2py 1 s1 2 ðs 1Þ2 0 Z 1 s=2 ða2 þ y 2 Þ arctanðy=aÞ cos½s arctanðy=aÞ þ2 : ð2:16Þ dy e2py 1 0 Calling the new integrals I1(s, a) and I2(s, a), respectively, we get I 1 ðs; aÞ ! 0; I 2 ðs; aÞ ! 2a
s ! 0; Z
1
dx 0
arctan x 1 1 þ Oða2 Þ ; ¼ e2pax 1 12a
s ! 0;
ð2:17Þ
and it follows that k 1 1 1X ð1Þ ð2kÞ!fð2k þ 2Þ : f0H ð0; aÞ a þ a log a þ 2kþ1 2 p k¼0 ð2paÞ
ð2:18Þ
Recall that the Barnes zeta function is written in terms of the Hurwitz zeta function, as given by the first of Eqs. (2.7) above. To obtain the derivative of the Barnes zeta function, however, we will proceed as before: we first work in terms of a general s in order to be able to analytically continue the expression to the desired Umgebung of s = 0, expand then in Taylor series around s = 0, and finally take the derivative at the point s = 0. Being more precise, we first expand fH(s, a) around s = 0, what yields fB ðs; aj~ rÞ ¼
rs r1 rs rs r2 1 fH ðs 1; a=r1 Þ þ 1 fH ðs; a=r1 Þ þ 1 sf ð1 þ s; a=r1 Þ þ Rðs; aj~ rÞ; s 1 r2 2 12 r1 H
ð2:19Þ
where the remainder term can be expressed in the following alternative ways (other possibilities may exist, too) "Z 8 2 # 1 1 > P arctan x dx 1 a r1 r1s > 1s 1 > ðn þ a=r1 Þ þ n þ Oðs2 Þ; > < ¼ 2s r2 exp½2pðr1 =r2 Þxðn þ a=r1 Þ 1 24 r2 r2 n¼0 0 Rðs; aj~ rÞ > 2kþ1 1 > srs P > r2 > ð1Þk ð2kÞ!fð2k þ 2Þ 2pr fH ðs þ 2k þ 1; a=r1 Þ þ Oðs2 Þ: : p1 1 k¼1
ð2:20Þ From here it is immediate to obtain, at s = 0, the formula (cf. Eq. (2.8)): r1 1 1 a r2 fB ð0; aj~ rÞ ¼ B2 ða=r1 Þ þ : þ 2 2 r1 2r2 12r1
ð2:21Þ
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
And from Eqs. (2.19), (2.20), taking the derivative with respect to s, at s = 0, we finally arrive to the very simple expression rÞ ¼ fB ð0; aj~ rÞ log r1 þ f0B ð0; aj~ where
Rðaj~ rÞ
r1 0 f ð1; a=r1 Þ þ Rðaj~ rÞ; r2 H
"Z 8 1 > P > 2r1 > ðn þ a=r1 Þ > < ¼ r2 > > > > : p1
r1 1 1 r2 B2 ða=r1 Þ þ log Cða=r1 Þ logð2pÞ wða=r1 Þ 2 4 2r2 12r1 ð2:22Þ
2 # arctan x dx 1 a r1 þ n ; exp ½2pðr1 =r2 Þxðn þ a=r1 Þ 1 24 r2 r2 n¼0 0 2kþ1 1 P k r2 ð1Þ ð2kÞ!fð2k þ 2Þ 2pr fH ð2k þ 1; a=r1 Þ: 1 1
ð2:23Þ
k¼1
Observe that, here, analyticity has been rigorously preserved at every step of the calculation. Those in Eq. (2.23) are two possible expressions for the remainder term Rða j ~ rÞ. The first of them is valid for 0 < r2 6 r1. It is very quickly convergent in this region and, therefore, very appropriate for numerical computation. Concerning this extreme, we note that, calling Z 1 1 X arctan x dx 1 ða þ bnÞ2 ; f ða; bÞ 2 ða þ bnÞ ð2:24Þ exp½2pða þ bnÞx 1 24 0 n¼0 one gets immediately (even with a pocket calculator) f ð1; 1Þ ¼ 0:0028;
f ð1; 10Þ ¼ 0:0023;
f ði; 1Þ ¼ 3:6 104 þ i0:084;
f ð10; 1Þ ¼ 1:5 105
f ð10i; 1Þ ¼ 2 106 þ i0:008;
f ð10; iÞ ¼ 3:3 105 þ i1:4 105 :
ð2:25Þ
Observe the very quick convergence of the integral expression, even for imaginary values of the argument. The second form of the remainder, Eq. (2.23), is an explicit asymptotic series for r2 2pr1. It coincides term by term with the asymptotic expansion obtained before. In fact, this second representation is exactly the same of the preceding subsection (cf. Eq. (2.12)). Notice once more that the initial symmetry under interchange of these two parameters has been explicitly broken by this assumption that permits the whole calculation to be carried out. Moreover, by treating apart (if necessary) a finite number of first terms in the original expression, it is easy to artificially increase the value of a and, with this, to considerably augment the speed and accuracy of the numerical results, in the first of expressions (2.23). 2.3. Analytic continuation to arbitrary values of r1, r2 Taking as starting point for the analytic continuation the formula: Z 1 1 1 X 1 X s ða þ r1 n1 þ r2 n2 Þ ¼ dt ts1 eðaþr1 n1 þr2 n2 Þt CðsÞ 0 n1 ;n2 ¼0 n1 ;n2 ¼0 Z 1 1 eat ; a; r1 ; r2 > 0; Re s > 2; ¼ dt ts1 CðsÞ 0 ð1 er1 t Þð1 er2 t Þ
ð2:26Þ
it is plain that it remains valid without change as long as the real parts of a, r1, r2 are all positive. It is also clear that in the complex planes corresponding to the parameters r1, r2, as soon as any of them gets purely imaginary, say r1 ¼ jr1 jeip=2 ¼ ijr1 j;
ð2:27Þ
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
than an infinite number of poles are created in the integrand, with X s1 1 X
1 eat 2pk s1 Re s t e2pka=jr1 j 1 e2pkr2 =jr1 j D1 ; ¼ r1 t Þð1 er2 t Þ jr ð1 e j 1 k¼1 r -poles
147
ð2:28Þ
1
which is an absolutely convergent series that will provide an additional contribution, 2pD1, in the analytic continuation process when, equivalently, the integration path for the integrand variable t (originally over the positive real axis) is deformed beyond the positive imaginary axis, in order to reach the possibility of setting Re r1 < 0 in the integral formula for the new integration path. Complementing this procedure—namely, just for one of the two parameters ri—with a formula for the analytic continuation of the Hurwitz zeta function fH(s, a) in the parameter a, we are done. This last can be read off from the usual tables, with rigorous prescriptions and domains of validity. Thus, the final formula for the analytic continuation in the parameter r1 has the form: Z 1 X 1 eas s þ 2pD1 ; Re a; Re r2 > 0; Re r1 < 0; ða þ r1 n1 þ r2 n2 Þ ¼ dsss1 r CðsÞ R ð1 e 1 s Þð1 er2 s Þ n ;n ¼0 1
2
ð2:29Þ being R a straight-line in the second quadrant of the r1-complex plane, starting at the origin. 2.4. Barnes’ related zeta function in two dimensions For the other, Barnes’ related, two-dimensional zeta function considered at the beginning, Eq. (2.3), we get 4 2 X X 1 a 1 a a a þ f0 ð0; aj~ rÞ ¼ f0B ð0;aj~ ra Þ þ log ri þ logðri Þ log C log C þ logð2pÞ ; 2 ri 2 ri ri ri a¼1 i¼1 ð2:30Þ where ~ r1 ¼ ðr1 ; r2 Þ;
~ r2 ¼ ðr1 ; r2 Þ;
~ r3 ¼ ðr1 ; r2 Þ;
~ r4 ¼ ðr1 ; r2 Þ:
ð2:31Þ
Note that this evaluation involves an analytic continuation on the values of r1 and r2—which are strictly positive in the case of the Barnes zeta function—to negative and, in general, complex values r1, r2 2 C. In this way we try to elude the unsurmountable problems that a direct interpretation of the sum in Eq. (2.3) poses. In particular, when r1 = r2 it develops an infinite number of zero and constant modes for an infinite number of possible values of the constants, what renders a direct interpretation of the series extremely problematic. After the analytic continuation performed here from the Barnes zeta function—together with a natural transportation of the negative signs of the indices ni to the parameters ri—the above limit is obtained in the final expressions, although with some severe restrictions (see below). Substituting the expressions (2.22) and (2.23) above into Eq. (2.30), after some long but straightforward computation, we obtain a a api a p 0 ð2:32Þ f ð0; aj~ rÞ ¼ log C log i þ log 2p: C þ r2 r2 r2 r2 2 Important considerations are here in order. (i) Observe, to begin with, that the last one is a finite expression and not an asymptotic series expansion as for the case of the Barnes zeta function. In fact, it has happened that the terms of the asymptotic expansions canceled one by one when performing the sum over all the ~ r 0 s. (ii) Second, note that the breaking of the initial symmetry under interchange of r1 and r2 in dealing with the analytic continuation (valid for r2 6 r1), has finally resulted in the disappearance of one of the two parameters from the end result. Remarkable is the fact, however, that the final formula actually preserves the modular invariance of the initial one, under the parameter change on the other variable, here r2 a ! a þ kr2 ;
k 2 Z;
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
since it transforms as f0 ð0; aj~ rÞ ! f0 ð0; aj~ rÞ þ 2pk; so that the modular symmetry of a with respect to r2 is preserved. What has happened to the r1 dependence? It has disappeared completely, the reason for this being the following. The formula for the Barnes zeta function for positive r1 and r2 obtained in the preceding section cannot be analytically continued to both r1 and r2 negative, because, for any value of s (and a, r1, r2 fixed), once we break the symmetry and do the analytic continuation say in r2, then the analytic continuation in r1 is restricted to the only possibility for n1 of being zero. In fact, in other words, when we try to pull out of the r1 real axis, for any value of n1 5 0 we encounter a pole of the initial zeta function, for Im r1 arbitrarily small—as these poles form a dense set in the complex r1plane—taking n2 big enough. Now, it is immediate to see that one can interpret the final formula (2.32) as the true analytic continuation for the restriction n1 = 0, namely: 1 X
f0 ð0; ajð0; r2 ÞÞ ¼ f0 ð0; ajð0; r2 ÞÞ ¼ ð2:32Þ:
ð2:33Þ
1
Let us quote also the following result, that we have obtained in two different ways: by analytically continuing in the factor 1 of n1 to 1 or, alternatively, on a and i to a and i, simultaneously. We get 1 1 X X
ða þ n1 þ in2 Þ
s
aið1 log aÞ
n1 ¼1 n2 ¼0
ipa p pa i ½wðaÞ wðaÞ 2 12 2 12
1 log½CðaÞCðaÞ log CðaiÞ 2 k 1 X ð1Þ B2kþ2 ½fH ð2k þ 1; aÞ fH ð2k þ 1; aÞ: þi ð2k þ 1Þð2k þ 2Þ k¼1
þ
ð2:34Þ
Finally, these formulas give us immediately the values of the determinants of the corresponding operators (say A) in each case [31–33]: rÞ ¼ exp f0A ð0; aj~ rÞ : detf Aðaj~ This issue will be further analyzed in the next section. 3. Barnes zeta functions in arbitrary dimensions There is no obstruction, in principle, to using the same procedure with general Barnes zeta functions in any dimension, d, and getting in this way corresponding formulas of the same type (involving multi-sums). The method will be described in this section. Using once more Hermite’s formula, we readily obtain the following recurrence for the Barnes zeta function in d dimensions ðdÞ
fB ðs; aj~ rÞ ¼
1 X
s
ða þ~ rd ~ nd Þ ;
Re s > d;
ð3:1Þ
0 ~ nd ¼~
in terms of corresponding ones in d 1 dimensions ðdÞ
fB ðs; aj~ rÞ ¼
r1 1 ðd1Þ ðd1Þ d fB ðs 1; aj~ rd1 Þ þ fB ðs; aj~ rd1 Þ 2 s1 Z 1 s=2 1 X ð1 þ x2 Þ sinðs arctan xÞ 1s 1 ða þ~ rd1 ~ nd1 Þ dx þ rd 1 2prd ðaþ~ rd1 ~ nd1 Þx e 1 0 ~ 0 n ¼~ d1
ð3:2Þ
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
149
Proceeding as in the section before, after some work we obtain the following analytic expansion of ðdÞ fB ðs; aj~ rÞ around s = 0: ðdÞ
fB ðs; aj~ rÞ ¼
½d=21 2kþ1 r1 1 ðd1Þ s X ðd1Þ k rd d fB ðs 1; aj~ rd1 Þ þ fB ðs; aj~ rd1 Þ þ ð1Þ 2 2p k¼0 s1 2p 1 X ðd1Þ 1s ð2kÞ!fð2k þ 2ÞfB ð2k þ s þ 1; aj~ rd1 Þ þ sr1 ða þ~ rd1 ~ nd1 Þ d ~ 0 nd1 ¼~
Z
1
dx 0
arctan x 1 ðaþ~ rd1 ~ nd1 Þx
e2prd
1
½d=21 X
ð1Þ
k¼0
k
r 2kþ2 d
2p
ð2kÞ!ða þ~ rd1 ~ nd1 Þ
# 2ðkþ1Þ
þ Oðs2 Þ: ð3:3Þ
Here [d/2 1] means integer part of d/2 1. The added and subtracted terms correspond to the poles of the d-dimensional Barnes zeta function. The convergence properties of the last term, i.e., the integral minus the subtracted terms, are as good for general d as for the case d = 2 (see the preceding section). 3.1. Derivatives of higher dimensional Barnes zeta functions We can now take the derivative at s = 0, with the following result: ðdÞ 0
1 ðd1Þ 0 ð1; aj~ rd1 Þ þ fB ð0; aj~ rd1 Þ 2 ½d=21 h i r 2kþ1 1 X d ðd1Þ þ ð1Þk ð2kÞ!fð2k þ 2ÞFP fB ð2k þ 1; aj~ rd1 Þ þ Rd ð0; aj~ rÞ; 2p k¼0 2p ðd1Þ
fB ð0; aj~ rÞ ¼ r1 d fB
ðd1Þ 0
ð1; aj~ rd1 Þ r1 d fB
ð3:4Þ
where FP[ ] means ‘finite part of’, and for the remainder term we have, as before, the two different expressions: "Z 8 1 1 P arctan x > > > ða þ~ rd1 ~ nd1 Þ dx 2pr1 ðaþ~r ~n Þx ¼ r1 > d > d1 d1 d > 1 e 0 ~ 0 nd1 ¼~ > > # > < ½d=21 2kþ2 P k rd 2ðkþ1Þ ð3:5Þ Rd ð0; aj~ rÞ ð1Þ ð2kÞ!ða þ~ rd1 ~ nd1 Þ ; > > 2p k¼0 > > > 2kþ1 > 1 > P > ðd1Þ k rd 1 > ð1Þ ð2kÞ!fð2k þ 2ÞfB ð2k þ 1; aj~ rd1 Þ: : 2p 2p k¼½d=2 ðdÞ 0
Observe that in order 0to obtain fB ð0; aj~ rÞ we need two derivatives of the d 1 zeta function, namely ðd1Þ 0 ðd1Þ fB ð0; aj~ rd1 Þ and fB ð1; aj~ rd1 Þ. The derivative at s = 1 can be calculated exactly in the same way as the one at s = 0, just by performing first the expansion of the Barnes zeta function around s = 1. For the case d = 2 (that is, for the first derivative needed in order to initiate the recurrence), the result is the following: r1 r1 0 ð2Þ0 ð2Þ fB ð1; aj~ rÞ ¼ r1 log r1 fB ð1; aj~ rÞ þ r1 f ð2; a=r1 Þ f ð2; a=r1 Þ 4r2 H 2r2 H 1 r2 r2 0 þ f0H ð1; a=r1 Þ f ð0; a=r1 Þ f ð0; a=r1 Þ þ R1 ð1; aj~ rÞ; ð3:6Þ 2 12r1 H 12r1 H the remainder term R1 ð1; aj~ rÞ being finite and given by the asymptotic expansion: 2kþ1 1 r1 X r2 k R1 ð1; aj~ rÞ ¼ ð1Þ ð2k 1Þ!fð2k þ 2Þ fH ð2k; a=r1 Þ; p k¼1 2pr1
ð3:7Þ
or by the corresponding expression as the difference of an integral and the term k = 0 of the series (same situation as above).
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E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
With this we close the circle and, we observe that this procedure allows for the calculation of any derivative of the Barnes zeta function in d dimensions in a recursive way and with a highly accurate numerical determination of the remainder terms. 3.2. Application to the multidimensional harmonic oscillator: determinants and the multiplicative anomaly (or defect) Let us consider the quantum harmonic oscillator in d dimensions, with angular frequencies (x1, . . . , xd). The eigenvalues read þ b; kn ¼ n x
n ðn1 ; . . . ; nd Þ;
ðx1 ; . . . ; xd Þ; x
b¼
d 1X xk 2 k¼1
ð3:8Þ
whose poles are to be found at the points and the related zeta function is the Barnes one, fd ðs; b j xÞ, s = k(k = d, d 1, . . . , 1). Their corresponding residua can be expressed in terms of generalized Bernoulli polyðdÞ defined by nomials Bdk ðbjxÞ, Qd
td eat
i¼1 ð1
ebi t Þ
1 ¼ Qd
1 X
i¼1 bi n¼0
n
BðdÞ n ðajbi Þ
ðtÞ : n!
ð3:9Þ
The residua of the Barnes zeta function are ¼ Re sfd ðk; bjxÞ
ð1Þ
dþk
ðk 1Þ!ðd kÞ!
Qd
j¼1 xj
ðdÞ
Bdk ðbjxÞ;
k ¼ d; d 1; . . . :
ð3:10Þ
At issue will be here the calculation, through the corresponding zeta function, of the determinant of the harmonic oscillator Hamiltonian in any number, d, of dimensions. As discussed before, in Quantum Field Theory the zeta function method is unchallenged at the one-loop level, where it is rigorously defined and where many calculations reduce basically to the computation of determinants. It is interesting to note, however, that for the determinants defined by means of the zeta function [31–33], detf A :¼ exp½f0A ð0Þ;
ð3:11Þ
the question of the non-commutative or multiplicative anomaly [28] appears (see, for instance [23,24]): the determinant of the product of two pseudodifferential operators (admitting both and their product a zeta function and corresponding zeta-determinants) is generically not equal to the product of the determinants. The logarithm of the quotient of the two results is termed the non-commutative (albeit it is actually non-zero in many cases where the two operators do commute, as we are going to see right now), or multiplicative anomaly.1 Using the formulas of the preceding sections, we are able to obtain the determinant of the harmonic oscillator Hamiltonian in d dimensions. We can then compare the result with the one obtained by making use of the celebrated formula of Wodzicki for the multiplicative anomaly: h i res ln Ab Ba 2 detf ðABÞ ; ð3:12Þ aðA; BÞ :¼ log ¼ detf ðAÞdetf ðBÞ 2abða þ bÞ where a > 0 and b > 0 are the orders of A and B respectively, while res is the Wodzicki or non-commutative residue (that is, the unique trace, up to multiplication by a constant, in the class of pseudodifferential operators) [37]. This formula is, in principle, not guaranteed to provide the right answer for the present situation, which involves non-compact operators. 1
Isadore Singer calls it a determinant defect. I thank him for illuminating discussions at his office at MIT, and for the very good tea.
E. Elizalde / Applied Mathematics and Computation 187 (2007) 141–152
151
Let V a constant potential, H the Hamiltonian operator corresponding to the free quantum harmonic oscillator in d dimensions, and as second operator let us take HV :¼ H + V (which actually commutes with H). For the multiplicative anomaly, we obtain ½d=2 ðdÞ 2k ð1Þd X ½c þ wðd 2kÞB2k ðbjxÞ aðH ; H V Þ ¼ Qd V : ð2kÞ!ðd 2kÞ! 2 j¼1 xj k¼1
ð3:13Þ
Here the generalized Bernoulli polynomials of odd order vanish and thus they can be dismissed from the sum. In spite of the manifold being non-compact, we are able to confirm the validity of the Wodzicki formula in the case under study, since exactly the same result, Eq. (3.13), is obtained by making use of Eq. (3.12). On the other hand, the remaining generalized Bernoulli polynomials are never zero, in fact d 1 X ðdÞ ¼ 1; BðdÞ B0 ðbjxÞ ðbj xÞ ¼ x2 ; 2 12 i¼1 i " # d X 1 7 X ðdÞ 4 2 2 ¼ x þ xi xj ; B4 ðbjxÞ ð3:14Þ 24 10 i¼1 i i<j " # d 5 31 X 7 X 4 2 X 2 2 2 ðdÞ 6 ¼ x þ xx þ x x x ; ... B6 ðbjxÞ 96 70 i¼1 i 10 i6¼j i j i<j
